Integrally closed domain

Definition

An integrally closed domain is a type of integral domain where every element that is integral over the domain actually lies within the domain itself. This means that if an element satisfies a monic polynomial with coefficients in the domain, then that element must be part of the domain. This concept is closely linked to prime and maximal ideals, since being integrally closed can help determine properties of these ideals and how they interact with the structure of the ring.

5 Must Know Facts For Your Next Test

1. Every field is an integrally closed domain because fields contain all their integral elements by definition.
2. A Dedekind domain is always integrally closed, but not all integrally closed domains are Dedekind domains.
3. If a one-dimensional Noetherian local ring is integrally closed, then it is a discrete valuation ring.
4. The property of being integrally closed can be used to determine whether certain prime ideals are maximal, which can simplify the study of the ring's structure.
5. Integrally closed domains are essential in algebraic geometry as they correspond to coordinate rings of integral schemes.

Review Questions

• How does the concept of integral elements relate to the definition of an integrally closed domain?
  • Integral elements are crucial to understanding integrally closed domains because these domains are defined by their relationship to such elements. If an element is integral over an integrally closed domain, it must actually belong to that domain. This shows how the integrity condition ensures closure; only elements that satisfy a polynomial with coefficients from the domain are considered, thereby highlighting the significance of these integral elements.
• Discuss how being integrally closed influences the properties of prime ideals in a ring.
  • Being integrally closed has a direct impact on the behavior of prime ideals within a ring. In particular, if an integral domain is integrally closed, it helps establish whether prime ideals are maximal. For example, in an integrally closed domain, any prime ideal contained in a maximal ideal may have additional properties due to closure under integral elements, thereby providing insights into how these ideals structure the overall ring.
• Evaluate the importance of integrally closed domains in algebraic geometry and how they connect to geometric objects.
  • Integrally closed domains play a vital role in algebraic geometry because they are linked to coordinate rings of algebraic varieties. When studying geometric objects, an integrally closed domain ensures that all necessary integral elements related to those varieties are contained within the ring itself. This characteristic is fundamental for defining schemes and studying morphisms between them. Additionally, it aids in ensuring certain desirable properties of varieties like irreducibility and separability, further establishing a deep connection between algebra and geometry.

Related terms

Integral Element:

An element in a ring that satisfies a polynomial equation with coefficients from a subring.

Noetherian Ring:

A ring in which every ascending chain of ideals stabilizes, which often helps in proving properties related to integrally closed domains.

Dedekind Domain:

A type of integral domain that is integrally closed, Noetherian, and has the property that every nonzero prime ideal factors uniquely into maximal ideals.